AN ALGEBRA OF PLANE PROJECTIVE GEOMETRY. 

 By H. B. Phillips and C. L. E. Moore. 



Presented by H. W. Tyler, February 14, 1912. Received January 29, 1912. 



Introduction. 



1. The Ausdehnungslehre of Grassmann ^ has been applied to prob- 

 lems in geometry in two ways. In the form of vector analysis it has 

 been used in solving problems of a metrical type. In the form of point 

 analysis (with homogeneous coordinates as a basis) it has been used in 

 problems of a descriptive nature. In projective geometry both of these 

 methods have certain advantages and also certain disadvantages. The 

 values of distances and angles occurring in vector analysis are useful as 

 variables in terms of which to express projective relations. Yet the fact 

 that these quantities are invariant under Euclidean motion has no place 

 in projective geometry. The ternary form of the point algebra is of 

 great value, but it is a decided disadvantage that XA and A (where A 

 is a point) though distinct are not descriptively distinguishable. It is 

 our aim in this paper to show how this coefficient A can be interpreted 

 as an angle invariant under a group of motions determined by the alge- 

 bra itself and thus while using the homogeneous form retain the essen- 

 tial advantages of the metrical system. 



"We first develop the two systems of analysis in a purely projective 

 way. Assuming that points and lines are represented by letters and that 

 addition follows the usual laws, we find that there must exist exceptional 

 elements somewhere in the plane. In fact, expressions AA, where A is a 

 point and A infinite, are not subject to the laws of addition. In the lan- 

 guage of coordinate geometry such points correspond to infinite values of 

 the coordinates and hence lie on a line. Likewise the lines Aa where a 

 is a line and A infinite are exceptional lines passing through a point. 

 Thus in such a system of algebra there exists a fixed point and a fixed 

 line. We define our additions relative to these. In the vector addition 

 the sum of two points A and B is a point C such that the harmonic of 

 the singular line with respect to C and the fixed point is the harmonic 

 of the same line with respect to A and B. This addition is characterized 



^ Gesammelte Werke, Vol. I, 1896. 



VOL. XLVII. — 47 



